A dose-finding design for phase I clinical trials based on Bayesian stochastic approximation

Background Current dose-finding designs for phase I clinical trials can correctly select the MTD in a range of 30–80% depending on various conditions based on a sample of 30 subjects. However, there is still an unmet need for efficiency and cost saving. Methods We propose a novel dose-finding design based on Bayesian stochastic approximation. The design features utilization of dose level information through local adaptive modelling and free assumption of toxicity probabilities and hyper-parameters. It allows a flexible target toxicity rate and varying cohort size. And we extend it to accommodate historical information via prior effective sample size. We compare the proposed design to some commonly used methods in terms of accuracy and safety by simulation. Results On average, our design can improve the percentage of correct selection to about 60% when the MTD resides at a early or middle position in the search domain and perform comparably to other competitive methods otherwise. A free online software package is provided to facilitate the application, where a simple decision tree for the design can be pre-printed beforehand. Conclusion The paper proposes a novel dose-finding design for phase I clinical trials. Applying the design to future cancer trials can greatly improve the efficiency, consequently save cost and shorten the development period. Supplementary Information The online version contains supplementary material available at 10.1186/s12874-022-01741-3.


.1 Transformation of dose levels
A general rule to transform the dose levels in (0, 1) is as follows.  If the original dose levels increase in fold change such as 100, 200, 400 and so on. Do logarithm transformation before Step 1. Figure S1 illustrates the approximation of π(x) in (v 0 , v 1 ) containing x n by a line segment. Proof. Our proof follows closely with Cheung [1]. (S1)

A.1.2 Illustration of the local modeling
Since ℓ m (θ, β) = F (x m ; θ, β)ℓ m−1 (θ, β) when y m = 1, the numerator of the integrant in (S1) can be expressed as Figure S2 shows the search paths by five other methods given the same responses of DLT as in Figure 1 for the proposed method. The search path by CRM deviates from the 7th cohort as (6, 6, 5, 6); the search path by mTPI deviates from the 7th cohort as (6, 6, 6, 6); the search pathes obtained by mTPI-2, BOIN and Keyboard deviate from the 8th cohort as (6, 6, 6). All these five methods lead to one level above the target (fifth) dose indicated by the horizontal line.    Table S1, where the action led by Wald interval is decomposed in escalation and de-escalation.

B.1.2 with historical information
We compare the proposed hBSA with iBOIN in the presence of historical information. The same specification of PESS is used by both methods. (The comparison with Hi3+3 is not included as it specifies PESS in a different way.) The iBOIN is carried out by free software at trialdesign.org.
In addition to the setup of the probabilities of toxicity in Table S1, two kinds of skeletons are specified in Tables S7-S8. The first kind matches with the true probabilities of DLT at MTD, called 'correctly specified skeleton'. The second kind mismatches with the true probabilities of DLT at MTD by one or two levels off (such as in Scenarios 2 and 1 respectively), called 'mis-specified skeleton', as described in A.3.2 of Zhou et al. [4].
Tables S9-S10 as well as Figures  Moreover, we repeat the comparison with the PESS doubled, i.e. n 0k = 6, to represent a prior with more historical data. So that when q k = 0.3, we have a k = 2 and b k = 4 proportionally. In parallel to Tables S9-S10, Tables S11-S12 report the comparison of the two competing methods in three metrics under four combinations made from target toxicity rate and skeleton specification with n 0k = 6. With more informative prior, the findings are consistent to those under the vague prior with PESS n 0k = 3. When the skeleton is correctly specified (Table S11), the superiority of hBSA to iBOIN is more pronounced. When the skeleton is mis-specified (  Figure S4 shows twenty random scenarios of toxicity rates for K = 5 and 6 under the target rate 30%. of the seven competing methods for K = 5 and 6 respectively, where the MTD equally probably located at the first four doses.

B.2.2 with historical information
We compare the proposed method with iBOIN in the presence of historical information under 200 random scenarios under K = 5 and 6, respectively, where the prior of MTDs are specified in the same two ways as in Section 3.1. For the mis-matched skeleton, 50 scenarios are randomly generated each for one or two levels off in two (opposite) directions. The complete results are given in Table S14.
When K = 5, the proposed method with fixed dose performs better than iBOIN in accuracy (w.r.t. both PCS and MTD%). The improvements are more pronounced by hBSA with exact dose information when the MTD is at early or middle position. The performance becomes inferior to iBOIN when the MTD resides at a late position, which is due to its conservatism noted before. These comparison results hold the same no matter the skeleton is correctly specified or mis-specified. When K = 6, hBSA outperforms iBOIN in PCS and MTD% under correctly specified skeleton. Under mis-specified case, hBSA performs inferior to iBOIN in MTD% when the MTD is at a late position. In all cases, hBSA yields significantly better overdose control than iBOIN regardless of the specification of the skeleton.
C Sensitivity analysis

C.1 Performance of BSA under different numbers of subintervals
Tables S15 and S16 report the performance of BSA using different number of subintervals s under the 20 representative scenarios in Table S1. It is seen that the performance are comparable to those under s = 3 (Tables S2-S5) with a slight variation between 1-2% in average PCS.

C.2 Performance of BSA under different cohort sizes
Tables S17 and S18 report the performance of BSA under the 20 representative scenarios in Table S1 with different number of cohort size. It is seen that the performance with smaller cohort sizes and varying cohort size in {1, 2, 3} improves in accuracy, especially for the scenarios where the MTD resides at a late position, and declines sensibly as necessary scarification in overdose control, which is still superior to the other competing methods in average (Tables S2-S5).
C.3 Performance of BSA when the MTD is randomly assigned at all K doses Table S19 reports the four metrics in comparison with the six competing methods as in Tables S13, where the MTD is randomly assigned at all K doses.  Table S20 reports the four metrics in comparison with the two competing methods as in Table S14 in the presence of historical information, where the MTD is randomly assigned at all K doses. Figure S3 shows an example of the online package at https://bsa4df.shinyapp.io/BSA app.

D Discussion
In this example, the user first inputs i) the target toxicity rate α = 0.3, ii) the number of doses K = 5, iii) the option how to use the dose level information, where 'rank' stands for the fixed dose method. Then, the user inputs the records for up to the latest cohort, i.e., dose level, cohort size, observed number of DLTs for each cohort. Note that varying number of cohort size is allowed.
After clicking the 'Generate Decision Tree' button, the right panel displays a decision tree for transition action for the next three cohorts assuming the cohort size is three. where the target DLT rates are 20% for the first ten scenarios and 30% for the last ten scenarios, respectively.